Syntax checks, as the name suggests, indicates any issues with the PDF syntax and acts like some sort of a master Profile to check syntax issues in your PDF. Switching it on to indicate such issues in the form of errors or mere warnings can count as a good practice.
How to enable/disable 'Syntax Checks'
On creating/editing a Profile, you can find a wide range of settings that are available across a number of categories like 'Document', 'Pages', 'Images', 'Colors' etc for a Profile, like shown below.
Under 'Document', you can find 'Report PDF syntax issues' amongst other Checks. Simply click on the dropdown next to it to enable/disable it, like shown below.